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# The Normal Distribution

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1. The Normal Distribution

2. The Normal Distribution • Distribution – any collection of scores, from either a sample or population • Can be displayed in any form, but is usually represented as a histogram • Normal Distribution – specific type of distribution that assumes a characteristic bell shape and is perfectly symmetrical

3. The Normal Distribution • Can provide us with information on likelihood of obtaining a given score • 60 people scored a 6 – 6/350 = .17 = 17% • 9 people scored a 1 – 3%

4. The Normal Distribution • Why is the Normal Distribution so important? • Almost all of the statistical tests that we will be covering (Z-Tests, T-Tests, ANOVA, etc.) throughout the course assume that the population distribution, that our sample is drawn from (but for the variable we are looking at), is normally distributed • Also, many variables that psychologists and health professionals look at are normally distributed • Why this is requires a detailed examination of the derivation of our statistics, that involves way more detail than you need to use the statistic.

5. The Normal Distribution • Ordinate • Density – what is measured on the ordinate (more on this in Ch. 7) • Abscissa

6. The Normal Distribution • Mathematically defined as: • Since  and e are constants, we only have to determine μ (the population mean) and σ (the population standard deviation) to graph the mathematical function of any variable we are interested in • Don’t worry, understanding this is not necessary to understanding the normal distribution, only a helpful aside for the mathematically inclined

7. The Normal Distribution • Using this formula, mathematicians have determined the probabilities of obtaining every score on a “standard normal distribution” (see Table E.10 in your book) • To determine these probabilities for the variable you’re interested in we must plug in your variable to the formula • Note: This assumes that your variable fits a normal distribution, if not, your results will be inaccurate

8. The Normal Distribution • However, this table refers to a Standard Normal Distribution • Μ = 0; σ = 1 • How do you get your variable to fit?

9. The Normal Distribution • Z-Scores • Range from +∞ to -∞ • Represent the number of standard deviations your score is from the mean • i.e. z = +1 is a score that is 1 standard deviation above the mean and z = -3 is a score 3 standard deviations below the mean • Now we can begin to use the table to determine the probability that our z score will occur using table E.10

10. The Normal Distribution

11. The Normal Distribution • Mean to Z

12. The Normal Distribution

13. The Normal Distribution • Larger Portion

14. The Normal Distribution • Smaller Portion

15. Reminder: Z-Scores represent # of standard deviations from the mean • For this distribution, if μ = 50 and σ = 10, what score does z = -3 represent? z = +2.5?

16. z = -.1 z = 1.645 (z = -.1, “Mean to Z”) + (z = 1.645, “Mean to Z”) .0398 + .4500 = .4898 = 49%

17. z = -1.00, “Smaller Portion” = Red + Blue • z = -1.645, “Smaller Portion” = Blue • (Red + Blue) - Blue = Red

18. z = -1.645 z = -1.00 (z = -1.00, “Smaller Portion”) – (z = -1.645, “Smaller Portion”) .1587 - .0500 = .1087 = 11%

19. The Normal Distribution • What are the scores that lie in the middle 50% of a distribution of scores with μ = 50 and σ = 10? • Look for “Smaller Portion” = .2500 on Table E.10 • z = .67 • Solve for X using z-score formula • Scores = 56.7 and 43.3

20. The Normal Distribution

21. The Normal Distribution • Other uses for z-scores: • Converting two variable to a standard metric • You took two exams, you got an 80 in Statistics and a 50 in Biology – you cannot say which one you did better in without knowing about the variability in scores in each • If the class average in Stats was a 90 and the s.d. 15, what would we conclude about your score now? How is it different than just using the score itself? • If the mean in Bio was a 30 and the s.d. was a 5, you did 4 s.d’s above the mean (a z-score of +4) or much better than everyone else

22. The Normal Distribution • Other uses for z-scores: • Converting variables to a standard metric • This also allows us to compare two scores on different metrics • i.e. two tests scored out of 100 = same metric one test out of 50 vs. one out of 100 = two different metrics • Is 20/50 better than 40/100? Is it better when compared to the class average? • Allows for quick comparisons between a score and the rest of the distribution it is a part of

23. The Normal Distribution • Standard Scores – scores with a predetermined mean and standard deviation, i.e. a z-score • Why convert to standard scores? • You can compare performance on two different tests with two different metrics • You can easily compute Percentile ranks • butthey are population-relative! • Percentile – the point below which a certain percent of scores fall • i.e. If you are at the 75th%ile (percentile), then 75% of the scores are at or below your score

24. The Normal Distribution • How do you compute %ile? • Convert your raw score into a z-score • Look at Table E.10, and find the “Smaller Portion” if your z-score is negative and the “Larger Portion” if it is positive • Multiply by 100

25. The Normal Distribution • New Score = New s.d. (z) + New Mean • New IQ Score = 15 (2) + 100 = 130 • T-Score – commonly used standardized normal distribution w/ mean = 50 and s.d. = 10 • T-Score = 10 (2) + 50 = 70